Modeling Backscatter Properties of Snowfall at Millimeter Wavelengths

MAY 2007 MATROSOV 1727

SERGEY Y. MATROSOV Cooperative Institute for Research in Environmental Sciences, University of Colorado, and NOAA/Earth System Research Laboratory, Boulder, Colorado

(Manuscript received 26 April 2006, in final form 20 July 2006)

ABSTRACT

Ground-based vertically pointing and airborne/spaceborne nadir-pointing millimeter-wavelength radars are being increasingly used worldwide. Though such radars are primarily designed for cloud remote sensing, they can also be used for precipitation measurements including snowfall estimates. In this study, modeling of snowfall radar properties is performed for the common frequencies of millimeter-wavelength radars such

as those used by the U.S. Department of Energy’s Atmospheric Radiation Measurement Program (Ka and W bands) and the CloudSat mission (W band). Realistic snowflake models including aggregates and single dendrite crystals were used. The model input included appropriate mass–size and terminal fall velocity–size relations and snowflake orientation and shape assumptions. It was shown that unlike in the Rayleigh scattering regime, which is often applicable for longer radar wavelengths, the spherical model does not generally satisfactorily describe scattering of larger snowflakes at millimeter wavelengths. This is especially true when, due to aerodynamic forcing, these snowflakes are oriented primarily with their major dimensions in the horizontal plane and the zenith/nadir radar pointing geometry is used. As a result of modeling using

the experimental snowflake size distributions, radar reflectivity–liquid equivalent snowfall rates (Ze–S) relations are suggested for “dry” snowfalls that consist of mostly unrimed snowflakes containing negligible amounts of liquid water. Owing to uncertainties in the model assumptions, these relations, which are

derived for the common Ka- and W-band radar frequencies, have significant variability in their coefficients that can exceed a factor of 2 or so. Modeling snowfall attenuation suggests that the attenuation effects in

“dry” snowfall can be neglected at the Ka band for most practical cases, while at the W band attenuation may need to be accounted for in heavier snowfalls observed at longer ranges.

1. Introduction As for rainfall, snowfall radar estimates are typically obtained with longer wavelength (centimeter) radars Traditional radar methods of measuring precipitation such as those that operate at S band (10–11-cm wave- are based on relations between water equivalent reflec- length, ␭), C band (␭ ϳ 5–6 cm), or X band (␭ ϳ 3 cm). tivity factor (hereafter “reflectivity”), Z , and precipi- e Snowfall rate as inferred from radar measurements is tation rate, R. Even for a simpler case of rainfall, there usually expressed in melted liquid equivalent, S, rather are significant variations in Z –R relations due to vari- e than in solid snow rate. Two different approaches are ability of the drop size distributions. Estimating snow- used to derive Z –S relations. In one approach both Z fall parameters from radar measurements is more chal- e e and S are calculated from theoretical or experimental lenging. Variability in snowflake shapes, densities, and snowflake size distributions (SSD) and they are related fall velocities results in greater uncertainties in Z –R e in a statistical manner (e.g., Sekhon and Srivastava relations for snowfall. Notwithstanding these uncertain- 1970; Matrosov 1992). In the second approach, radar ties, radar gives an opportunity to obtain near-real-time measurements of reflectivity are related to the snowfall estimates of snowfall rate and accumulation, providing rate as observed by nearby snow gauges (e.g., Super information that is not readily available with other re- and Holroyd 1998; Hunter and Holroyd 2002). Rasmus- mote sensors. sen et al. (2003) provides a review of some Ze–S relations suggested for use with longer-wavelength radars. Corresponding author address: Dr. Sergey Matrosov, R/PSD2, For such radars, snowflakes are typically much smaller 325 Broadway, Boulder, CO 80305. than radar wavelength and scattering does not deviate E-mail: [email protected] significantly from the Rayleigh law.

DOI: 10.1175/JAS3904.1

JAS3904 1728 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 64

At millimeter radar wavelengths such as those at Ka 2. Snowflake model band (␭ ϳ 8 mm) and, especially, W band (␭ ϳ 3 mm), a. Snowflake shape scattering by typical snowflakes is generally non- Rayleigh. Radars operating at these wavelengths have The Rayleigh scattering conditions at a radar wave- been primarily intended for applications in cloud re- length ␭ for a snow particle with a major dimension D mote sensing and they have typically not been used for can be formulated as follows: snowfall measurements. It has been shown, however, x ϭ ␲Dր␭ K 1, ͑1͒ that the use of such radars in pair with centimeter- wavelength radars can potentially improve the accuracy ␲ | |ր␭ K ͑ ͒ D ms 1, 2 of snowfall parameter retrievals (Matrosov 1998; Liao et al. 2005; Wang et al. 2005) because the dual wave- where ms is the complex refractive index of snow. The length reflectivity ratio provides independent informa- great majority of derivations of Ze–S relations in snow- tion on characteristic size of snowflakes. Knowing size fall assumed that for snowflakes that satisfy these con- information, one can better infer snowfall rate from ditions the radar backscatter would be the same as that single reflectivity measurements. of a sphere of solid ice having the same mass. While this Currently, a number of vertically pointing millime- assumption precludes modeling polarimetric properties ter-wavelength radars operate on a continuing basis in of backscatter cross sections, it provides a reasonably many locations including those that regularly or some- good approximation of Ze for low-density dry snow at times get snowfalls. Examples of such radars are verti- centimeter wavelengths. The spherical model, however, becomes progressively less suitable as the size param- cally pointing K -band (34.6 GHz) millimeter- a eter x increases beyond the bounds of the Rayleigh wavelength cloud radars (MMCR) operated by the U.S. conditions. Larger nonspherical ice particles exhibit Department of Energy’s Atmospheric Radiation Mea- backscatter dependence on particle orientation (e.g., surement Program (DOE ARM) in Oklahoma and the Matrosov 1993; Aydin and Singh 2004) and generally North Slope of Alaska (NSA). Another MMCR-type greater reflectivities (especially outside the Rayleigh radar has been recently deployed in the Canadian Arc- scattering regime) compared to those of spherical par- tic. The W-band (94 GHz) nadir-looking CloudSat ra- ticles of the same mass (Matrosov et al. 2005a). dar will provide global information on reflectivity pro- The simplest geometrical shape that allows describ- files of hydrometeors (Stephens et al. 2002). Although ing nonspherical hydrometeors is that of spheroid. In the main objective of all these radars is cloud remote their classical observational in situ study, Magono and sensing, their measurements of snowfall can also be Nakamura (1965) used tracing outlines of snowflakes interpreted quantitatively, thus providing valuable in- that revealed a generally spheroidal shape. The oblate formation on snowfall accumulation and the vertical spheroidal model is used widely in the radar community structure of snowfall rate, which can vary significantly to model raindrops and it was also shown to satisfacto- (Mitchell 1988). rily describe and explain depolarization observations of Since for most of the radars mentioned above there snowflakes at Ka band (Matrosov et al. 1996, 2001). are no collocated centimeter-wavelength radars for ap- Note also that, at radar frequencies, the small details in plication of dual-wavelength ratio techniques, Ze–S re- a particle structure usually do not significantly affect lations specifically tailored for millimeter wavelengths the backscatter properties that depend largely on the are needed for quantitative interpretation of snowfall overall shape (Dungey and Bohren 1993), which, in the measurements. This article presents results of modeling case of spheroid, is determined by the spheroid aspect Ze–S relations for the common millimeter-wavelength ratio, r. This is especially true for smaller snowflake size frequencies of 34.6 and 94 GHz based on available in- parameters x ϭ ␲D␭Ϫ1 (Schneider and Stephens 1995). formation on snowflake-size distributions and densities. The applicability of the spheroid model for larger snow- The first of the aforementioned approaches for deriving flakes with small values of r (e.g., dendrites that are

Ze–S relations is employed here because there are no larger than several millimeters) at W band is not so reliable datasets available that combine collocated mil- obvious. This model, however, satisfactorily explains limeter-wavelength radar and snow gauge measure- experimental 94-GHz aggregate snow reflectivities ments. This modeling is concerned with so called “dry” (Matrosov et al. 2005a). snow, the term usually used for describing snow with Based on the arguments given above, and also due to negligible amounts of liquid water/riming. “Dry” snow- the fact that very large dendrites are not very common, falls usually contribute most significantly to snow accu- the oblate spheroidal model was adopted here for mulation at temperatures less than about Ϫ8°C. snowflake modeling. Another reason to use this model MAY 2007 MATROSOV 1729 is the lack of detailed shape information on aggregates, A lower limit of D ϭ 0.01 cm was chosen in (3a) be- which makes the input information for the computa- cause for smaller sizes this formula can provide results tional models that are able to account for fine shape that correspond to unrealistic densities exceeding those details (e.g., the discrete dipole approximation; Lemke of solid ice. While (3b) closely approximates data from and Quante 1999) not very well defined. In the spher- Magono and Nakamura (1965), (3a) provides particle oid model, low aspect ratio values of about 0.1–0.3 are masses that are within the range of empirical relations suitable for modeling pristine dendrite snow crystals, of the aforementioned studies for smaller aggregate while values of r Ϸ 0.5–0.7 are usually observed for particles. Note that (3a) is very close to the relation for larger aggregate crystals (Magono and Nakamura 1965; aggregates S3 (according to the Magono and Lee 1966 Korolev and Isaac 2003). The value around r Ϸ 0.6 also classification) suggested by Mitchell (1996): best explains experimental dual-wavelength ratios from ϭ 2.1 ͑ ͒ ͑ ͒ the airborne radars operating at 9.6 and 94 GHz (Ma- m 0.0028D cgs units , 4 trosov et al. 2005a). and is also in good agreement with later findings by Mitchell and Heymsfield (2005). b. Snowflake orientation Equation (3c) reflects the fact that the density of very Magono and Nakamura (1965) noted that, due to large snowflakes does not exhibit any significant depen- aerodynamic forcing, snowflakes typically fall with dence on size. their major dimensions oriented horizontally. Polari- In addition to aggregate snowflakes, modeling was metric radar depolarization measurements at different also performed for single dendrite snowflakes. While polarimetric bases and at different elevation angles also snowfalls consisting of single dendrites typically do not reveal that, in the absence of strong horizontal winds yield significant accumulations on the ground, such (and wind shear), single shape snowflakes are oriented modeling was performed in order to better understand preferably horizontally with a standard deviation (SD) the generality of the derived Ze–S relations. According of about 9° (Matrosov et al. 2005b). Thus it was as- to Heymsfield (1972) density–size relations for den- sumed during modeling that symmetry axes of oblate drites (P1e according to the Magono and Lee classifi- spheroids that approximate snowflakes are oriented cation) and stellars with broad arms (P1d) are practi- with the 0° mean and SD ϭ 9° with respect to vertical. cally identical. For such crystals Mitchell (1996) sug- While there is no polarization dependence of radar gests the following relation: measurables for vertically pointing radar for such snow- 1.67 flake orientations, snowflake shape and, to a lesser ex- m ͑g͒ ϭ 0.000 27D ͑0.009 cm Ͻ D Յ 0.15 cm͒, tent, the value of SD influence the magnitude of Ze at ͑5͒ millimeter-wavelength frequencies. which was used for modeling of dentritic crystals in this c. Snowflake mass study. According to Magono and Nakamura (1965), density, ␳, of falling dry aggregate snowflakes that are larger d. Snowflake fall velocity than 0.2 cm diminishes with their size from approxi- ␷ Terminal fall velocities of hydrometeors, t, are de- Ϫ3 ϭ Ϫ3 mately 0.03 g cm at D 0.2 cm to about 0.01 g cm termined by their mass, their area projected to the flow, ϭ Ϸ at D 1.5 cm. While D 0.2 cm is the smallest size their size, and such air properties as density ␳, and dy- presented by Magono and Nakamura (1965), there are namic viscosity ␩. In a recent study, Mitchell and a number of microphysical studies in which ice aggre- Heymsfield (2005) critically addressed existing fall ve- gates with sizes less than 0.2 cm were studied (e.g., locity–size relations for aggregate ice crystals. They Brown and Francis 1995; Mitchell 1996; Heymsfield et suggested a refined relation that accounts for the cor- al. 2004). Most of these studies indicate that the mass, respondence between the Reynolds and the Best or b m, of an aggregate particle typically increases as D , Davies numbers that is appropriate for aggregates. The Ϸ where b 2. The following formulas provide a con- data presented by Mitchell and Heymsfield (2005, their tinuous mass–size relation for snowflakes for a large Fig. 5) indicate an approximately linear relation be- range of possible sizes: ␷ tween t and the logarithm of aggregate size. This linear m ͑g͒ ϭ 0.003D2 ͑0.01 cm Ͻ D Յ 0.2 cm͒, ͑3a͒ relation can be expressed as ͑ ͒ ϭ 2.5 ͑ Ͻ Յ ͒ ͑ ͒ ␷ ͑ Ϫ1͒ Ϸ ϩ ͓ ͑ ͒ Ϫ ͔͑ Ͼ ␮ ͒ m g 0.0067D 0.2 cm D 2.0 cm , 3b t cm s 30 50 log10 D 2 for D 100 m , m ͑g͒ ϭ 0.0047D3 ͑D Ͼ 2cm͒. ͑3c͒ ͑6͒ 1730 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 64

where D is in microns. For 0.16 cm Ͻ D Ͻ 1 cm, expression (6) also approximates well the measurement- derived fall velocity relation for large unrimed aggregates of dendritic crystals suggested by Locatelli and Hobbs (1974). Many theoretical (e.g., Mitchell and Heymsfield 2005) and experimental (e.g., Magono and Nakamura 1965) studies indicate that terminal fall velocities of larger snowflakes practically do not depend upon the size of the flake. Taking this into consideration, it was assumed that (6) describes aggregate snowflake fall velocities in the range 0.01 cm Ͻ D Ͻ 1 cm and the ␷ Ն constant value of t was then assumed for D 1 cm. For single dentritic snowflakes (P1d), the fall velocity–size relation can be approximated as

␷ ϭ 0.3 ͑ Ͻ ͒ ͑ ͒ t 75D P1d, D 0.15 cm, cgs units , 7 which is in general agreement with the Heymsfield (1972), Mitchell (1996), and Mitchell and Heymsfield (2005) data. Terminal fall velocities aloft at an altitude h can be calculated from those at the surface using the expression (Pruppacher and Klett 1997)

␷ ͑ ͒ ϭ ␷ ͑ ͓͒␳͑ ͒ր␳͑ ͔͒␣Ϫ1ր͓␩͑ ͒ր␩͑ ͔͒1Ϫ2␣ ͑ ͒ t h t 0 h 0 h 0 , 8 where ␩ and ␳ are the air dynamic viscosity and density [␳(0) ϭ 1.23 kg mϪ3, ␩(0) ϭ 1.65 ϫ 10Ϫ5 Pa s], respectively, and ␣ ϭ 0.57 for snowflakes. It should be noted that the fall velocity approximations considered above are applicable to snowflakes with a negligible degree of riming. Rimed and wet snowflakes and also graupel FIG. 1. Backscatter cross sections of horizontally oriented ag- generally exhibit higher fall velocities. Snowfalls con- gregate snowflakes with aspect ratios 0.6, 0.8, and 1.0 at vertical incidence and frequencies (a) 34.6 and (b) 94 GHz as a function of sisting of such particles are usually observed at tem- snowflake major dimension. peratures near freezing and when significant amounts of supercooled water drops are present in the atmosphere. This study focuses on “dry” snow. erties of dry snow at radar frequencies (e.g., Battan 1973; Oguchi 1983):

͑ 2 Ϫ ͒͑ 2 ϩ ͒Ϫ1 Ϸ ͑ 2 Ϫ ͒͑ 2 ϩ ͒Ϫ1 ͑ ͒ 3. Backscatter cross sections of individual ms 1 ms 2 Pi mi 1 mi 2 , 9 snowflakes where mi is the complex refractive index of solid ice The T-matrix method (Barber and Yeh 1975) was adopted from Warren (1984). ␴ used for calculations of snowflake scattering properties. Figure 1 shows backscatter cross sections, b, of hori- Input parameters for the T-matrix code include snow- zontally oriented aggregate snowflakes with aspect ra- flake size, aspect ratio, complex refractive index and tios that are in the range 0.6 Յ r Յ 1 when viewed at the direction of the incident radar beam. A snowflake is vertical incidence. The mass of snowflakes was mod- considered a uniform mixture of solid ice and air where eled according to (3) and was preserved as r was chang- solid ice details that are typically much smaller than the ing. This means that effective bulk densities are differ- radar wavelength are ignored. Based on the mass and ent for snowflakes with the same size but different as- volume of a snowflake, its effective density is calculated pect ratios. Though Mie-type backscatter oscillations

and the relative volumes of solid ice (Pi) and air (Pa) are present for all considered aspect ratios, as can be are computed. The complex refractive index of a bulk seen from Fig. 1, backscatter cross sections of non-

snowflake mixture, ms, is then calculated using the ap- spherical snowflakes already begin to deviate notice- proach that is often used in modeling dielectric prop- ably from the spherical model already at major dimen- MAY 2007 MATROSOV 1731

FIG. 3. Backscatter cross sections of horizontally oriented for dendritic crystals (P1d) as a function of crystal major dimension.

Figure 3 shows backscatter cross sections for single pristine dendritic crystals (P1d) as a function of their size. Though the calculations presented in this figure were made for r ϭ 0.2 they are expected also to be representative for smaller values of r because of the ␴ lesser sensitivity of b to r at smaller values of aspect ratios and also because pristine crystals usually are smaller than aggregates and they are not expected to exceed a few millimeters in size. Note also that because the effective bulk density of snowflakes changes with their size, the results presented in Figs. 1–3 for one FIG. 2. As in Fig. 1 but for aspect ratios 0.2, 0.3, and 0.5. frequency cannot be obtained exactly from the results for the other frequency using simple frequency scaling. sions of about D ϭ 0.15 cm for ␯ ϭ 34.6 GHz, and D ϭ ␯ ϭ 0.05 cm for 94 GHz. This indicates that, for at least 4. Reflectivity of ensembles of snowflakes vertical-viewing geometry and millimeter radar wavelengths, the assumption that the backscatter cross sec- a. Snowflake size distributions tion of an arbitrary snowflake would be the same as Compared to the raindrop size distributions, infor- that of a sphere of ice having the same mass is not mation on snowflake size distributions (SSDs) in terms generally valid. This assumption, which is often used of real (not melted) snowflake dimensions is relatively when calculating radar properties of snowflakes is, scarce. However, different experimental and modeling however, valid for lower radar frequencies in the Ray- studies (e.g., Braham 1990; Harimaya et al. 2000; Senn leigh scattering regime. and Barthazy 2004) indicate that SSDs are generally Ϸ While, as mentioned above, r 0.6 is expected to exponential in shape down to a size of 1 mm, or even adequately describe natural snowflake aggregates, it is less. Since snowflakes with sizes greater than 1 mm also instructive to examine backscatter cross sections typically dominate in both radar reflectivity and snow- for particles with smaller aspect ratios. Figure 2 pre- fall rate, the exponential model for snowflake number Յ Յ sents results for 0.2 r 0.5. Note that the T-matrix concentration in a form calculation routine often becomes unstable for particles with aspect ratios smaller than about 0.2, so no results ͑ ͒ ϭ ͑Ϫ⌳ ͒ ͑ ͒ N D No exp D 10 are presented for r Ͻ 0.2. It can be seen from Fig. 2 that ␴ the dependence of backscatter cross sections b on r can be considered a reasonable modeling assumption. becomes progressively less pronounced as r decreases. One of the most complete experimental studies of This is true especially for smaller particles. snowflake size spectra in terms of snowflake real sizes 1732 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 64

FIG. 4. Scatterplot of the intercept and slope parameters in experimental snowflake size distributions.

was presented by Braham (1990) who reported data that contained 49 experimental SSDs. Figure 4 shows these SSDs approximated by the exponential function ⌳ with parameters No and from Braham (1990). The majority of snowflakes in Braham’s study appeared to be aggregate snowflakes with some amount of single spatial dendrites. Heavier snowfalls are characterized by smaller values of the parameter ⌳. Later experimental studies (e.g., Harimaya et al. 2000) showed experimental SSD features similar to those reported by Bra- ham (1990).

FIG. 5. Scatterplots of radar reflectivity at vertical incidence and snowfall rate at frequencies (a) 34.6 and (b) 94 GHz for experi- b. Ze–S relations at millimeter wavelengths mental snowflake size distributions. Reflectivity of a snowflake ensemble is given by the integral ␳ where w is the density of water, and the mass–size and fall velocity–size relations were discussed in sections 2c Dmax Ϫ ϭ ␭4␲ 5 |͑ 2 ϩ ͒ր͑ 2 Ϫ ͒| 2͵ ͗␴ ͘ ͑ ͒ and 2d. The values of Dmin and Dmax in (10) and (11) Ze m 2 m 1 b N D dD, Dmin were assumed to be 0.01 and 1 cm. This choice of Dmin and D proved to be not very sensitive for the con- ͑11͒ max sidered experimental SSDs. Modest variations of Dmin and D (about 2%–20%) in the integral (12) did not where m is the complex refractive index of water at a max result in significant changes in either Z or S. given wavelength, ␭, and the angular brackets in (11) e Figure 5 shows the scatterplots of radar reflectivity indicate additional integrating over the snowflake ori- versus liquid equivalent snowfall rate for frequencies entations, which were assumed to be Gaussian with a 0° 34.6 and 94 GHz calculated for the experimental SSDs mean and SD ϭ 9° as discussed in section 2b. Snow- with parameters from Fig. 4. Aggregate snowflake as- flake orientations in the azimuthal direction were aspect ratio was assumed to be 0.6 and the mass and sumed to be random, so for vertical incidence, Z does e terminal fall velocities were calculated using (3) and not depend on the polarization state of radar signals. (6). The best power-law fits drawn with Z as an inde- Liquid equivalent snowfall rate, S, can be expressed as e pendent variable for the aggregate snowflake assump- ϭ 1.20 ϭ 0.8 D tion are Ze 56S (34.6 GHz) and Ze 10S (94 max Ϫ Ϫ Ϫ1 6 3 1 ϭ ␳ ͵ ͑ ͒␷ ͑ ͒ ͑ ͒ ͑ ͒ GHz), where Ze in mm m and S in mm h . Note that S w m D t D N D dD, 12 Dmin first the S–Ze relations were developed considering Ze MAY 2007 MATROSOV 1733

6 Ϫ3 Ϫ1 TABLE 1. Variability of Ze–S (Ze in mm m , S in mm h ) relations due to main assumptions about snowflake mass m, terminal ␷ fall velocity t, and aspect ratio r.

Ze–S relations for the aspect ratio r ϭ 0.6, and Changing r Increasing/decreasing Increasing/decreasing ␷ → ␷ (3), (6) used for m–D, t–D 0.6 0.8 m by 20% t by 20% ϭ 1.2 ϭ 1.1 ϭ 1.2 ϭ 1.2 ϭ 1.2 ϭ 1.2 34.6 GHz Ze 56S Ze 34S Ze 66S /Ze 48S Ze 46S /Ze 67S ϭ 0.8 ϭ 0.8 ϭ 0.8 ϭ 0.8 ϭ 0.8 ϭ 0.8 94 GHz Ze 10S Ze 6S Ze 13S /Ze 8S Ze 8S /Ze 12S

as an independent variable and then these relations (6) and an assumption about the r ϭ 0.6 aspect ratio

were converted to the conventional Ze–S form. The were used to produce the data presented in Fig. 5. data scatter around the best-fit lines is rather modest. Table 1 shows how the best fit power-law Ze–S relations Due to stronger non-Rayleigh scattering effects, reflec- would vary if one of the main assumptions is changed. tivities at 94 GHz are significantly smaller than those at To assess this variability the particle mass and fall ve- 34.6 GHz for the same values of snowfall rate S. The locity were changed by 20% compared to values from

millimeter wavelength Ze–S relations are quite differ- Eqs. (3) and (6), respectively, and the particle aspect ent from those obtained by various researches for ratio was changed from 0.6 to 0.8. A more random longer wavelength radars (Rasmussen et al. 2003). orientation of snowflakes will be manifested similarly While it is expected that aggregate snowflakes are as an increase in r. These changes in the main assump- the dominant hydrometeor type in appreciable snow- tions may represent the reasonable uncertainties of cor- falls, for comparison reasons, calculations were also responding assumptions. Indeed, the fall velocity ap- performed using the dendritic crystal model and the proach of Mitchell and Heymsfield (2005), for example, same experimental SSDs. The mass–size and fall veloci- provides fall velocities of 1 cm aggregates (which is a ty–size relations in this case are given by (5) and (7). typical aggregated snowflake size) of about 1 m sϪ1. The results for which dendrite reflectivities were The Khvorostyanov and Curry (2002) approach pro- greater than 0 dBZ are also shown in Fig. 5. For 34.6 vides 1.4 m sϪ1 for the same particle size. This is about GHz the aggregate best fit also rather well describes the Ϯ20% from their mean value of 1.2 m sϪ1. Two differ- dendrite data. It is not quite so for the 94-GHz data, ent mass–size relations for aggregates from Mitchell although deviations of the dendrite data from the best (1996) provide similar percentage differences if applied power-law fit for aggregates are still not very signifi- to larger aggregate sizes. cant. Reflectivities that are less than about 0 dBZ are It can be seen from Table 1 that there is an appre-

more characteristic of nonprecipitating clouds and were ciable sensitivity of the Ze–S relations to main assump- not considered here. tions. Assuming the independence of the assumption uncertainties, one can expect variability of at least a factor of 2 in the coefficients of the Z –S relations. c. Variability of Ze–S relations due to assumptions e Variability in the exponents of these relations is more The main assumptions affecting radar reflectivity val- modest, though it also contributes to the overall uncer- ues, Ze, include those about the particle mass and tainty in the correspondence between Ze and S. It shape. While in the Rayleigh scattering regime, back- should be noted also that, compared to the low obser- scatter cross sections of individual particles are propor- vational angles, the vertical radar beam observations tional to the square of particle masses, the resonance (which is the most common configuration for Ka- and effects diminish the backscatter–mass dependence. The W-band radars) and the preferable orientation of snow- shape effects, which are not very pronounced for small flakes with their major dimension in the horizontal particles, however, become progressively more impor- plane result in larger variability of backscatter due to tant (especially for vertical incidence at horizontally changes in the particle aspect ratio. oriented scatterers) as particles grow larger and transi- tion to the sizes that are outside the Rayleigh scattering regime. The assumption about particle mass also affects 5. Attenuation of radar signals in snowfall the liquid equivalent snowfall rates, S. The values of S are also affected by the assumption about the terminal Attenuation of radar signals at centimeter wave- fall velocity–size relation. lengths in dry snowfall is very small and is usually ne- The aggregate particle mass–size relation given by glected. The T-matrix approach used in this study to (3) and the terminal fall velocity–size relation given by calculate backscatter properties of snowflakes was also 1734 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 64

band, however, is about one order of magnitude higher

than that at Ka band and, probably, needs to be accounted for when measuring snowfall with higher precipitation rates in thick snow layers. The total attenuation in such cases can be appreciable and reach or even exceed 1 dB, which is on the order of a typical absolute calibration uncertainty of radars. For wet and/ or rimed snow, attenuation can be significantly larger. It should be noted also that the attenuation in the atmospheric gases (i.e., water vapor and oxygen) generally should be accounted for when interpreting measurements at W band at longer ranges (Stephens et al.

2002). Gases attenuation at Ka band at vertical incidence that typically amounts to fractions of 1 dB often can be neglected especially in dry cold atmospheres. FIG. 6. Scatterplots of the attenuation coefficient at vertical incidence and snowfall rate for experimental snowflake size distributions. 6. Summary and conclusions

Millimeter-wavelength radars operating at Ka and W used to estimate attenuation of Ka- and W-band signals bands are being increasingly used in the atmospheric in snowfall. The specific attenuation coefficient, a, can remote sensing. These radars are typically employed be obtained by integrating imaginary parts of the for- either in a vertical-pointing mode (for ground based ward scattering amplitudes f: radars) or in a nadir-pointing mode (for airborne and spaceborne radars). A number of ground-based milli- D max meter wavelength cloud profiling radars have been de- a ϭ 2␭͵ ͗Im͓f͑D͔͒͘N͑D͒ dD, ͑13͒ Dmin ployed in the colder climates and spaceborne radars are usually designed to operate at polar orbits, thus provid- where the angular brackets are used, as before, to in- ing frequent coverage of high latitudes where snowfall dicate additional integrating over the snowflake orien- is common. Although these radars’ primary use is re- tations. As was the case for reflectivity, there is no po- mote sensing of nonprecipitating clouds, they can also larization dependence in the attenuation coefficient for potentially be used for snowfall measurements. Adding the vertical incidence and mean snowflake orientation quantitative snowfall information derived from radar with major dimensions in the horizontal plane. The for- measurements will be useful in many climate and cloud ward scattering amplitude f relates to the dimensionless physics applications. amplitude matrix element So, which customarily is used In this study, modeling of reflectivity–liquid- in optics as equivalent snowfall rate was performed for the com- ϳ mon frequencies employed at Ka band ( 35 GHz) and ϭϪ ␲ ␭Ϫ1 ͑ ͒ ϳ So 2 if . 14 W band ( 94 GHz). “Dry” snowfall, which is characterized by negligible amounts of liquid water in snow Figure 6 shows scatterplots of the attenuation coef- and snowflake riming, was studied. Nonspherical aggre- ficients at Ϫ5°C versus liquid-equivalent snowfall rates gate and single crystal dendritic snowflake models were at Ka- and W-band radar frequencies calculated for the used for calculations. These models included realistic assumed snowflake aspect ratio r ϭ 0.6 and the same mass–size and terminal fall velocity–size relations and experimental SSDs that were used in reflectivity mod- assumed some flutter around the preferred orientation eling. Complex refractive indices of solid ice corre- of snowflakes with their maximum dimensions in the sponding to a temperature of Ϫ5°C are about 1.78 horizontal plane. Experimental size distributions de- Ϫi0.0024 (34.6 GHz) and 1.78 Ϫi0.0043 (94 GHz). scribing real (not melted) snowflake sizes were em- The best-fit power law approximations for the data in ployed in this study. Fig. 6 are a ϭ 0.011S1.1 and a ϭ 0.12S1.1 for 34.6 and 94 Most common aggregate snowflake sizes are already

GHz, respectively. It can be seen that the dry snow outside the Rayleigh scattering regime at Ka- and es- attenuation at Ka band is fairly small and can be ne- pecially at W-band frequencies. For vertical/nadir- glected for most practical cases. The attenuation at W pointing geometry, backscatter properties of snow- MAY 2007 MATROSOV 1735 flakes noticeably depend on snowflake overall shape REFERENCES (i.e., aspect ratio). The typical radar reflectivity– Aydin, K., and J. Singh, 2004: Cloud ice crystal classification using snowfall rate relations for aggregate snowflakes calcu- a 95-GHz polarimetric radar. J. Atmos. Oceanic Technol., 21, lated for a set of basic assumptions (see Table 1) are 1679–1688. ϭ 1.20 ϭ 0.8 Ze 56 (34.6 GHz) and Ze 10S (94 GHz). These Barber, P., and C. Yeh, 1975: Scattering of electromagnetic waves relations are generally valid also for snowfall consisting by arbitrarily shaped dielectric bodies. Appl. Opt., 14, 2864– of single dendrite crystals (P1d), though snowfall rates 2872. Battan, L. J., 1973: Radar Observation of the Atmosphere. Uni- and thus accumulations of such snowfalls are usually versity of Chicago Press, 324 pp. significantly less compared to the snowfalls consisting Braham, R. R., Jr., 1990: Snow particle size spectra in lake effect of aggregates. Though these relations were developed snows. J. Appl. Meteor., 29, 200–207. for “dry” snowfall, they are expected to be applicable Brown, P. R. A., and P. N. Francis, 1995: Improved measurements for snowfall with some light degree of riming. Dry snow of the ice water content in cirrus using a total-water probe. J. typically forms at colder temperatures (usually less than Atmos. Oceanic Technol., 12, 410–414. Ϫ Dungey, C. E., and C. F. Bohren, 1993: Backscattering by non- about 12°C) and is characterized by mean vertical spherical hydrometeors as calculated by the coupled-dipole Doppler velocities that are less than about 1.2–1.4 method: An application in radar meteorology. J. Atmos. Oce- Ϫ ms 1. A significant increase in snowflake riming and/or anic Technol., 10, 526–532. wetness is likely to cause changes in Ze–S relations. Harimaya, T., H. Ishida, and K. Muramoto, 2000: Characteristics The strong non-Rayleigh behavior of backscatter at of snowflake size distributions connected with the difference millimeter wavelengths in K and W bands results in of formation mechanism. J. Meteor. Soc. Japan, 78, 233–239. a Heymsfield, A. J., 1972: Ice crystal terminal velocities. J. Atmos. reflectivity being proportional to the lower moment of Sci., 29, 1348–1357. the snowflake size distribution. As a result the expo- ——, A. Bansemer, C. Schmitt, C. Twohy, and M. R. Poellot, nents in the Ze–S relations are closer to unity compared 2004: Effective ice particle densities derived from aircraft to the longer wavelength radars. The proportionality of data. J. Atmos. Sci., 61, 982–1003. Hunter, S. M., and E. W. Holroyd, 2002: Demonstration of im- Ze and S to similar moments of the size distribution can, in part, explain small data scatter in Fig. 5. For precipi- proved operational water resources management through use of better snow water equivalent information. Bureau of Rec- tation radar frequencies (3–10 GHz), both coefficients lamation Rep. R-02-02, U.S. Department of the Interior, 75 and exponents in Ze–S relations are larger. Typically for pp. such frequencies, coefficients vary from 50 to 400 while Khvorostyanov, V. I., and J. A. Curry, 2002: Terminal velocities of the exponents vary between 1.5 and 2 (e.g., Rasmussen droplets and crystals: Power laws with continuous parameters et al. 2003). over the size spectrum. J. Atmos. Sci., 59, 1872–1884. Korolev, A., and G. Isaac, 2003: Roundness and aspect ratio of Uncertainties in the coefficients of Z –S relations e particles in ice clouds. J. Atmos. Sci., 60, 1795–1808. caused by variability in the snowflake shape, and in the Lemke, H. M., and M. Quante, 1999: Backscatter characteristics parameters of mass–size and terminal fall velocity–size of nonspherical ice crystals: Assessing the potential for po- relations are rather substantial. These coefficient un- larimetric radar measurements. J. Geophys. Res., 104, 31 739– certainties can easily be as high as a factor of ϳ2; thus, 31 752. the errors in the snowfall rate and accumulation esti- Liao, L., R. Meneghini, T. Iguchi, and A. Detwiler, 2005: Use of dual-wavelength radar for snow parameter estimates. J. At- mations for radar reflectivity measurements could be mos. Oceanic Technol., 22, 1494–1506. appreciable. These errors can potentially be reduced if Locatelli, J. D., and P. Hobbs, 1974: Fall speeds and masses of the coefficient tuning is performed based on collocated solid precipitation particles. J. Geophys. Res., 79, 2185–2197. radar and snow gauge measurements. Such a tuning Magono, C., and T. Nakamura, 1965: Aerodynamic studies of approach is often used in rainfall measurements when falling snowflakes. J. Meteor. Soc. Japan, 43, 139–147. ——, and C. W. Lee, 1966: Meteorological classification of natural parameters of standard Ze–R (R here is rainfall rate) snow crystals. J. Fac. Sci., Hokkaido Univ., Ser. 2, 7, 321-335. relations are adjusted based on the rain gauge measure- Matrosov, S. Y., 1992: Radar reflectivity of snowfall. IEEE Trans. ments that are deployed in the radar coverage area. Geosci. Remote Sens., 30, 454–461. There are, however, no high quality collocated datasets ——, 1993: Possibilities of cirrus particle sizing from dual- for millimeter-wavelength radars and snow gauge data frequency radar measurements. J. Geophys. Res., 98, 20 675– available at the moment, so this direction for refine- 20 684. ——, 1998: A dual-wavelength radar method to measure snowfall ment of Ze–S relations should be left for further studies. rate. J. Appl. Meteor., 37, 1510–1521. Acknowledgments. This research was supported by ——, R. F. Reinking, R. A. Kropfli, and B. W. Bartram, 1996: Estimation of ice hydrometeor types and shapes from radar the Office of Science (Biological and Environmental polarization measurements. J. Atmos. Oceanic Technol., 13, Research), U.S. Department of Energy, Grant DE- 85–96. FG02-05ER63954, and the CloudSat project. ——, ——, ——, B. E. Martner, and B. W. Bartram, 2001: On the 1736 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 64

use of radar depolarization ratios for estimating shapes of ice time correlation of radar reflectivity with snow gauge accu- hydrometeors in winter clouds. J. Appl. Meteor., 40, 479–490. mulation. J. Appl. Meteor., 42, 20–36. ——, A. J. Heymsfield, and Z. Wang, 2005a: Dual-frequency ra- Schneider, T. L., and G. L. Stephens, 1995: Theoretical aspects of dar ratio of nonspherical atmospheric hydrometeors. Geo- modeling backscattering by cirrus ice particles at millimeter phys. Res. Lett., 32, L13816, doi:10.1029/2005GL023210. wavelengths. J. Atmos. Sci., 52, 4367–4385. ——, R. F. Reinking, and I. V. Djalalova, 2005b: Inferring fall Sekhon, R. S., and R. C. Srivastava, 1970: Snow size spectra and attitudes of pristine dendritic crystals from polarimetric radar radar reflectivity. J. Atmos. Sci., 27, 299–307. data. J. Atmos. Sci., 241–250. 62, Senn, P., and E. Barthazy, 2004: In-situ observations and model- Mitchell, D. L., 1988: Evolution of snow-size spectra in cyclonic ing of aggregation of snowfall. Proc. European Conf. on Ra- storms. Part I: Snow growth by vapor deposition and aggre- dar Meteorology (ERAD-2004), Visby, Sweden, Copernicus gation. J. Atmos. Sci., 3431–3451. 45, GmbH, 253–256. ——, 1996: Use of mass- and area-dimensional power laws for Stephens, G. L., and Coauthors, 2002: The CloudSat mission and determining precipitation particle terminal velocities. J. At- the A-train. Bull. Amer. Meteor. Soc., 83, 1771–1790. mos. Sci., 53, 1710–1723. ——, and A. J. Heymsfield, 2005: Refinements in the treatment of Super, A. B., and E. W. Holroyd, 1998: Snow accumulation algo- ice particle terminal velocities, highlighting aggregates. J. At- rithm for the WSR-88D radar: Final report. Bureau of Rec- mos. Sci., 62, 1637–1644. lamation Rep. R-98-5, U.S. Department of the Interior, 75 Oguchi, T., 1983: Electromagnetic wave propagation and scatter- pp. ing in rain and other hydrometeors. Proc. IEEE, 71, 1029– Wang, Z., G. M. Heymsfield, L. Li, and A. J. Heymsfield, 2005: 1078. Retrieving optically thick ice cloud microphysical properties Pruppacher, H. R., and J. D. Klett, 1997: Microphysics of Clouds by using airborne dual-wavelength radar measurements. J. and Precipitation. Kluwer Academic, 954 pp. Geophys. Res., 110, D19201, doi:10.1029/2005JD005969. Rasmussen, R., M. Dixon, S. Vasiloff, F. Hage, S. Knight, J. Vive- Warren, S. G., 1984: Optical constants of ice from the ultraviolet kanandan, and M. Xu, 2003: Snow nowcasting using a real- to the microwave. Appl. Opt., 23, 1206–1225.